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Cesàro average in short intervals for Goldbach numbers


Authors: Alessandro Languasco and Alessandro Zaccagnini
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 11P32; Secondary 11P55
DOI: https://doi.org/10.1090/proc/13645
Published electronically: May 4, 2017
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Abstract: Let $ \Lambda $ be the von Mangoldt function and

$\displaystyle R(n) = \sum _{h+k=n} \Lambda (h)\Lambda (k).$    

Let further $ N,H$ be two integers, $ N\ge 2$, $ 1\le H \le N$, and assume that the Riemann Hypothesis holds. Then

$\displaystyle \sum _{n=N-H}^{N+H} R(n) \Bigl (1- \frac {\vert n- N \vert }{H}\Bigr )$ $\displaystyle = HN -\frac {2}{H} \sum _{\rho } \frac {(N+H)^{\rho +2} - 2 N^{\rho +2} +(N-H)^{\rho +2} }{\rho (\rho + 1)(\rho + 2)}$    
  $\displaystyle + \mathcal {O}\Bigl ({N \Bigl (\log \frac {2N}{H}\Bigr )^2 + H (\log N)^2 \log (2H) }\ ,$    

where $ \rho =1/2+i\gamma $ runs over the non-trivial zeros of the Riemann zeta function $ \zeta (s)$.

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Additional Information

Alessandro Languasco
Affiliation: Dipartimento di Matematica “Tullio Levi-Civita”, Università di Padova, Via Trieste 63, 35121 Padova, Italy
Email: languasco@math.unipd.it

Alessandro Zaccagnini
Affiliation: Dipartimento di Matematica e Informatica, Università di Parma, Parco Area delle Scienze 53/a, 43124 Parma, Italy
Email: alessandro.zaccagnini@unipr.it

DOI: https://doi.org/10.1090/proc/13645
Keywords: Goldbach-type theorems, Hardy-Littlewood method
Received by editor(s): June 1, 2016
Received by editor(s) in revised form: November 1, 2016
Published electronically: May 4, 2017
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2017 American Mathematical Society